2-5: Determinants And Multiplicative Inverses Of Matrices

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2-5The termdeterminant isoften used to meanthe value of thedeterminant.onAp Evaluatedeterminants. Find inverses ofmatrices. Solve systemsof equations byusing inversesof matrices.l WorealdOBJECTIVESRDeterminants and MultiplicativeInverses of Matricesp li c a tiThis situation can be described by a system of equations represented by amatrix. You can solve the system by writing and solving a matrix equation.Each square matrix has a determinant. The determinant ofnumber denoted by 87 46 is a8 48 4 7 6 or det 7 6 . The value of a second-orderdeterminant is defined as follows. A matrix that has a nonzero determinant iscalled nonsingular.Second-OrderDeterminantExampleMarshall plans to invest 10,500 into two differentbonds in order to spread out his risk. The first bond has an annualreturn of 10%, and the second bond has an annual return of 6%. IfMarshall expects an 8.5% return from the two bonds, how much shouldhe invest into each bond? This problem will be solved in Example 5.INVESTMENTSThe value of det1 Find the value of aa12 b1b2 , or aa12 b1, is a1b2 a2b1.b2 7 6 .84 7 6 8(6) 7(4) or 208 4The minor of an element of any nth-order determinant is a determinant oforder (n 1). This minor can be found by deleting the row and columncontaining the element. a1 b1 c1a2 b2 c2a3 b3 c3The minor of a1 is b c .b2 c2One method of evaluating an nth-orderdeterminant is expanding the determinant by minors.The first step is choosing a row, any row, in the matrix.At each position in the row, multiply the element timesits minor times its position sign, and then add theresults together for the whole row. The position signsin a matrix are alternating positives and negatives,beginning with a positive in the first row, first column.98Chapter 2Systems of Linear Equations and Inequalities33 .

Example a1 b1 c1a2 b2 c2 a1a3 b3 c3Third-OrderDeterminant2 Find the value of 45 2c2c3 6 14 b1 a2a3c2c3 c1 a2a3b2b3 23 . 3 4 62 13535 15 13 4 ( 6) 24 3 2 3 24 24 3GraphingCalculatorTip 4( 9) 6( 9) 2(18) 18You can use the det(option in the MATHlistings of the MATRXmenu to find adeterminant.For any m mmatrix, the identitymatrix, 1, mustalso be an m mmatrix. b2b3The identity matrix for multiplication for any square matrix A is the matrix I,such that IA A and AI A. A second-order matrix can be represented by1 01 0a1 b1a b1a1 b1a b1. Since 1 1, the matrix0 10 1a2 b2a2 b2a2 b2a2 b2 10 01 is the identity matrix for multiplication for any second-order matrix.IdentityMatrix forMultiplicationThe identity matrix of nth order, In, is the square matrix whose elements inthe main diagonal, from upper left to lower right, are 1s, while all otherelements are 0s.Multiplicative inverses exist for some matrices. Suppose A is equal toa1 b1a2 b2 , a nonzero matrix of second order. The term inversematrix generallyimplies themultiplicativeinverse of amatrix. xThe inverse matrix A 1 can be designated as x12 y1y2 . The product of amatrix A and its inverse A 1 must equal the identity matrix, I, for multiplication. aa12a1x1 b1 x2 a x b x2 12 2 a y b y1 0a y b y 0 1 b1b2 xx12y11 0y2 0 11 11 22 12 2From the previous matrix equation, two systems of linear equations can bewritten as follows.a1x1 b1x2 1a1y1 b1y2 0a2 x1 b2 x2 0a2y1 b2y2 1Lesson 2-5Determinants and Multiplicative Inverses of Matrices99

By solving each system of equations, values for x1, x2, y1, and y2 can be obtained. ba1b2 a2b1a1y2 a1b2 a2b1ba1b2 a2b1 a2x2 a1b2 a2b12x1 If a matrix A has adeterminant of 0then A 1 does notexist.Inverse of aSecond-OrderMatrix1y1 The denominator a1b2 a2b1 is equal to the determinant of A. If thedeterminant of A is not equal to 0, the inverse exists and can be defined asfollows. abIf A a1 b1 and22a1 b1a2 b2 0, then A 1 1a1 b1a2 b2 b a 22 b1.a1 A A 1 A 1 A I, where I is the identity matrix.ExampleGraphingCalculatorAppendixFor keystroke instruction on how to find theinverse of a matrix, seepages A16-A17.3 Find the inverse of the matrixFirst, find the determinant of 24 24 34 . 3.42 3 2(4) 4( 3) or 2044 4120 4The inverse is 1 531or 25 3 201 10Check to see if A A 1 A 1 A 1.Just as you can use the multiplicative inverse of 3 to solve 3x 27, you canuse a matrix inverse to solve a matrix equation in the form AX B. To solve thisequation for X, multiply each side of the equation by the inverse of A. When youmultiply each side of a matrix equation by the same number or matrix, be sure toplace the number or matrix on the left or on the right on each side of the equation tomaintain equality. B A 1B A 1B A 1BAXA 1AXIXXExampleMultiply each side of the equation by A 1.A 1 A IIX X4 Solve the system of equations by using matrix equations.2x 3y 17x y 4Write the system as a matrix equation.23x 17 1 1y4 To solve the matrix equation, first find the inverse of the coefficient matrix. 100Chapter 21231 1 1 1 31 1 3 5 122Systems of Linear Equations and Inequalities 21 3 2( 1) (1)(3) or 5 1

Now multiply each side of the matrix equation by the inverse and solve.23x 171 1 31 1 3 5 15 121 1y24 xy 1 5 The solution is ( 1, 5).l WoreaAponldRExamplep li c a ti5 INVESTMENTS Refer to theapplication at the beginning of thelesson. How should Marshall dividehis 10,500 investment between thebond with a 10% annual return anda bond with a 6% annual return sothat he has a combined annualreturn on his investments of 8.5%?First, let x represent the amount toinvest in the bond with an annualreturn of 10%, and let y represent theamount to invest in the bond with a6% annual return. So, x y 10,500since Marshall is investing 10,500.Write an equation in standard form that represents the amounts invested inboth bonds and the combined annual return of 8.5%. That is, the amount ofinterest earned from the two bonds is the same as if the total were invested ina bond that earns 8.5%.10%x 6%y 8.5%(x y)0.10x 0.06y 0.085(x y)0.10x 0.06y 0.085x 0.085y0.015x 0.025y 03x 5y 0Interest on 10% bond 10%xInterest on 6% bond 6%yDistributive PropertyMultiply by 200 to simplify the coefficients.Now solve the system of equations x y 10,500 and 3x 5y 0. Write thesystem as a matrix equation and solve.x y 10,5003x 5y 0 131 5 131 5Multiply each side of 1 5 18 31the equation by theinverse of thecoefficient matrix. xy 10,5000 xy 18 5 3 10,5000 11 xy 6562.53937.5 The solution is (6562.5, 3937.5). So, Marshall should invest 6562.50 in thebond with a 10% annual return and 3937.50 in the bond with a 6% annualreturn.Lesson 2-5Determinants and Multiplicative Inverses of Matrices101

C HECKU N D E R S TA N D I N GFORRead and study the lesson to answer each question.1. Describe the types of matrices that are considered to be nonsingular. 34 2 0does not have a determinant. Give another 3 5example of a matrix that does not have a determinant.2. Explain why the matrix3. Describe the identity matrix under multiplication for a fourth-order matrix.4. Write an explanation as to how you can decide whether the system ofequations, ax cy e and bx dy f, has a solution.Guided PracticeFind the value of each determinant.4 135. 27. 4105 15 1 2107 15 8. Find the inverse of each matrix, if it exists.9. 25 37 12 26326.10.6 4 10 33 9 00 46 69 Solve each system of equations by using a matrix equation.11. 5x 4y 312. 6x 3y 63 3x 5y 245x 9y 8513. MetallurgyAluminum alloy is used in airplane construction because it isstrong and lightweight. A metallurgist wants to make 20 kilograms of aluminumalloy with 70% aluminum by using two metals with 55% and 80% aluminumcontent. How much of each metal should she use?E XERCISESPracticeFind the value of each determinant.A 2 5 3 414. 2 3 2 1 17. 426. Find det A if A 10218. 513 16. 12 19. 7 8 2 130 21 3021. 367 2 411123. 3 4 1 22121320. 0 4 10 115.22. 2536 15 12 21715924. 3125. 9 1216 650 8 8 9 33 5 7 1 2 4 1.5 3.6 2.2 1.68.2 6.601 4323 .8 34 Chapter 2 Systems of Linear Equations and Inequalities www.amc.glencoe.com/self check quiz

Find the inverse of each matrix, if it exists.B 22 3 2 6 730. 6 7 21 00 4631. 8 12 27. 41 22 9 1332. 27 36 28.33. What is the inverse of29. 3 451 81 ? 2Solve each system by using a matrix equation.34. 4x y 135. 9x 6y 1236. x 5y 2637. 4x 8y 738. 3x 5y 2439. 9x 3y 1x 2y 74x 6y 123x 3y 03x 2y 415x 4y 35x y 1Solve each matrix equation. The inverse of the coefficient matrix is given.C40.41.GraphingCalculator 3 23122 21 1 6539 2 1311x 4 4 1 101y 0 , if the inverse is 9 3 3 3 .z15 18 x 9 1 211y 5 , if the inverse is 9 12 1521 .z 11521 33 Use a graphing calculator to find the value of each determinant. 2 4 2 323 6042.09 4 54 7 1843. 2 91 10 1204 66 14 1151 3 84701 8032 1Use the algebraic methods you learned in this lesson and a graphing calculator tosolve each system of equations.44. 0.3x 0.5y 4.7445. x 2y z 712x 6.5y 1.2l WoreaAponldRApplicationsand ProblemSolvingp li c a ti6x 2y 2z 44x 6y 4z 1446. IndustryThe Flat Rock auto assemblyplant in Detroit, Michigan, producesthree different makes of automobiles. In1994 and 1995, the plant constructed atotal of 390,000 cars. If 90,000 more carswere made in 1994 than in 1995, howmany cars were made in each year?47. Critical ThinkingDemonstratethat the expression for A 1 is themultiplicative inverse of A for anynonsingular second-order matrix.48. ChemistryHow many gallons of 10% alcohol solution and 25% alcohol solutionshould be combined to make 12 gallons of a 15% alcohol solution?Lesson 2-5 Determinants and Multiplicative Inverses of Matrices103

49. Critical ThinkingIf A ac bd , does (A )2 1 (A 1)2? Explain.50. GeometryThe area of a triangle with vertices at (a, b), (c, d), and (e, f ) cana b 11be determined using the equation A c d 1 . What is the area of a2e f 1triangle with vertices at (1, 3), (0, 4), and (3, 0)? (Hint: You may need to use theabsolute value of the determinant to avoid a negative area.) 51. RetailSuppose that on the first day of asale, a store sold 38 complete computersystems and 53 printers. During thesecond day, 22 complete systems and44 printers were sold. On day three ofthe sale, the store sold 21 systems and26 printers. Total sales for these items forthe three days were 49,109, 31,614, and 26,353 respectively. What was the unitcost of each of these two selected items?52. EducationThe following type of problem often appears on placement tests orcollege entrance exams.Jessi has a total of 179 points on her last two history tests. The secondtest score is a 7-point improvement from the first score. What are herscores for the two tests?Mixed Review53. GeometryThe vertices of a square are H(8, 5), I(4, 1), J(0, 5), and K(4, 9). Usematrices to determine the coordinates of the square translated 3 units left and4 units up. (Lesson 2-4)8 7354. Multiplyby . (Lesson 2-3)4 40 55. Solve the system x 3y 2z 6, 4x y z 8, and 7x 5y 4z 10.(Lesson 2-2)56. Graph g(x) 2 x 5 . (Lesson 1-7)57. Write the standard form of the equation of the line that is perpendicular toy 2x 5 and passes through the point at (2, 5). (Lesson 1-5)58. Write the point-slope form of the equation of the line that passes through thepoints at (1, 5) and (2, 3). Then write the equation in slope-intercept form.(Lesson 1-4)59. SafetyIn 1990, the Americans with Disabilities Act (ADA) went into effect.This act made provisions for public places to be accessible to all individuals,regardless of their physical challenges. One of the provisions of the ADA is thatramps should not be steeper than a rise of 1 foot for every 12 feet of horizontaldistance. (Lesson 1-3)a. What is the slope of such a ramp?b. What would be the maximum height of a ramp 18 feet long?60. Find [f g](x) and [g f](x) if f(x) x2 3x 2 and g(x) x 1. (Lesson 1-2)104Chapter 2 Systems of Linear Equations and InequalitiesExtra Practice See p. A29.

61. Determine if the set of points whose coordinates are (2, 3), ( 3, 4), (6, 3),(2, 4), and ( 3, 3) represent a function. Explain. (Lesson 1-1)62. SAT Practice The radius of circle E is 3.BSquare ABCD is inscribed in circle E. Whatis the best approximation for the differencebetween the circumference of circle E andthe perimeter of square ABCD?A3B2C1D 0.5E0CEDACAREER CHOICESAgricultural ManagerWhen you hear theword agriculture, youmay think of a quaintlittle farmhousewith chickens andcows runningaround like in thestorybooks of yourchildhood, buttoday OldMcDonald’s farm is bigbusiness. Agriculturalmanagers guide and assistfarmers and ranchers in maximizingtheir profits by overseeing the day-to-dayactivities. Their duties are as varied asthere are types of farms and ranches.An agricultural manager may oversee oneaspect of the farm, as in feeding livestockon a large dairy farm, or tackle all of theactivities on a smaller farm. They also mayhire and supervise workers and overseethe purchase and maintenance of farmequipment essential to the farm’s operation.CAREER OVERVIEWDegree Preferred:Bachelor’s degree in agricultureRelated Courses:mathematics, science, financeOutlook:number of jobs expected to declinethrough 2006Number of Farms and AverageFarm Size 1975–1998Farms(million)3.00Acresper Farm500United 51.801.6035019751980198519901995Source: NASS, Livestock & Economics BranchFor more information on careers in agriculture, visit: www.amc.glencoe.comLesson 2-5 Determinants and Multiplicative Inverses of Matrices105

GRAPHING CALCULATOR EXPLORATION2-5B Augmented Matricesand ReducedRow-Echelon FormAn Extension of Lesson 2-5OBJECTIVE Find reducedrow-echelonform of anaugmentedmatrix to solvesystems ofequations.Each equation isalways writtenwith the constantterm on the right.Another way to use matrices to solve a system of equations is to use an augmentedmatrix. An augmented matrix is composed of columns representing the coefficients ofeach variable and the constant term.Identify the coefficientsand constants.1x 2y 1z 73x 1y 1z 22x 3y 2z 7system of equationsx 2y z 73x y z 22x 3y 2z 7 1 21 731 1 2232 7Through a series of calculations that simulate the elimination methods you used inalgebraically solving a system in multiple unknowns, you can find the reduced 1 0 0 c1 row-echelon form of the matrix, which is 0 1 0 c2 , where c1, c2, and c30 0 1 c3A line is oftendrawn to separatethe constantscolumn.augmented matrixrepresent constants. The graphing calculator has a function rref( that will calculatethis form once you have entered the augmented matrix. It is located in the MATHsubmenu when MATRX menu is accessed.For example, if the augmented matrix above isstored as matrix A, you would enter the matrixname after the parenthesis and then insert aclosing parenthesis before pressing ENTER .The result is shown at the right.Use the following exercises to discover howthis matrix is related to the solution ofthe system.TRY THESEWrite an augmented matrix for each system of equations. Then find thereduced row-echelon form.1. 2x y 2z 7x 2y 5z 14x y z 1WHAT DO YOUTHINK?2. x y z 6 02x 3y 4z 3 04x 8y 4z 12 03. w x y z 02w x y z 1 w x y z 02x y 04. Write the equations represented by each reduced row-echelon form of thematrix in Exercises 1-3. How do these equations related to the original system?5. What would you expect to see on the graphing calculator screen if theconstants were irrational or repeating decimals?106Chapter 2 Systems of Linear Equations and Inequalities

The identity matrix for multiplication for any square matrix A is the matrix I, such that IA A and AI A . A second-order matrix can be represented by . Since , the matrix is the identity matrix for multiplication for any second-order matrix. Multiplicative

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